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Evaluating Definite Integrals for AP Calculus

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus

In this study guide:

  • Definite Integrals Involving Algebraic Functions
  • Definite Integrals Involving Absolute Value
  • Definite Integrals Involving Trigonometric,  Logarithmic, and Exponential Functions
  • Definite Integrals Involving Odd and Even Functions

Definite Integrals Involving Algebraic Functions

Example 1

Verify your result with a calculator.

Example 2

Verify your result with a calculator.

Example 3

Verify your result with a calculator.

Definite Integrals Involving Absolute Value

Example 1

Verify your result with a calculator.

Example 2

Verify your result with a calculator.

Definite Integrals Involving Trigonometric, Logarithmic, and Exponential Functions

Example 1

Verify your result with a calculator.

Example 2

Verify your result with a calculator.

Example 3

Verify your result with a calculator.

Example 4

Verify your result with a calculator.

Definite Integrals Involving Odd and Even Functions

If f is an even function, that is, f (–x) = f (x), and is continuous on [–a, a], then

If f is an odd function, that is, F (x) = – f (–x), and is continuous on [–a, a] then

Example 1

Evaluate

Since f (x) = cos x is an even function,

Verify your result with a calculator.

Example 2

Evaluate

Since f (x)=x4 – x2 is an even function, i.e., f (–x ) = f (x), thus

Verify your result with a calculator.

Example 3

Evaluate

Since f (x) = sin x is an odd function, i.e., f (–x)= – f (x), thus

You can also verify the result with a calculator.

Example 4

If for all values of k, then which of the following could be the graph of f ? See Figure 11.3-1.

Definite Integrals Involving Odd and Even Functions

Thus f is an even function. Choice (C).

Practice problems for these concepts can be found at: Definite Integrals Practice Problems for AP Calculus

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